Vacuum pressure swing adsorption system for N₂/CO₂ separation in consideration of unstable feed concentration

Abstract

In this study, an efficient PID control system has been designed for a vacuum pressure swing adsorption (VPSA) process which used silica gel to capture CO₂ from dry flue gas. Changes of CO₂ composition in feed gas are set as disturbances and vary from 12.0 to 15.0% to make simulation work more closely to reality. Meanwhile, adsorption step time duration was set to change consistently according to the feedback of CO₂ purity in product. PID control strategy built in software gPROMS is employed in this paper. The competence of the control system is firstly analyzed by a transient impulse of CO₂ content in feed gas (15.0% dropped to 12.0%). Then random pulse is added to test the ability of the control system to reject the unknown disturbances and track the set point. Detailed changes in bed under open and closed-loop are listed and made comparisons. Results demonstrate that the model shows an enhanced performance in the presence of random disturbances under closed-loop control compared with the open-loop operation. And this proves that closed-loop feedback PID control can be used to improve the CO₂ capture of VPSA process operations.

Appendix: Mathematical model

Assumptions for the whole model are listed as follows:

• 1. Gas phase obey the role of ideal gas.

• 2. Gas concentration, temperature and pressure only vary in axial direction.

• 3. Pressure drop along the bed is calculated by Ergun equation.

• 4. Thermal balance is maintained between gas and solid phase.

• 5. Kinetics of adsorption abided by linear driving force model.

• 6. Porosity of both bed and adsorbent particle is uniform along the bed.

The equations used in this paper are listed in follows:

Parameter	Expression
Component mass balance	$- \frac{\partial }{{\partial z}}\left( {{\varepsilon _b}{D_{ax}}c\frac{{\partial {y_i}}}{{\partial z}}} \right)+\frac{{\partial \left( {{v_g}{c_i}} \right)}}{{\partial z}}+\left( {{\varepsilon _b}+\left( {1 - {\varepsilon _b}} \right){\varepsilon _p}} \right)\frac{{\partial {c_i}}}{{\partial t}}+{\rho _P}\left( {1 - {\varepsilon _b}} \right)\frac{{\partial {q_i}}}{{\partial t}}=0$
Overall mass balance	$- \frac{\partial }{{\partial z}}\left( {{\varepsilon _b}{D_{ax}}\frac{{\partial c}}{{\partial z}}} \right)+\frac{{\partial \left( {{v_g}c} \right)}}{{\partial z}}+\left( {{\varepsilon _b}+\left( {1 - {\varepsilon _b}} \right){\varepsilon _p}} \right)\frac{{\partial c}}{{\partial t}}+{\rho _P}\left( {1 - {\varepsilon _b}} \right)\frac{{\partial q}}{{\partial t}}=0$
Energy balance	$\begin{gathered} \left\{ {\left[ {{\varepsilon _b}+\left( {1 - {\varepsilon _b}} \right){\varepsilon _p}} \right]\mathop \sum \limits_{{i=1}}^{N} {c_i}\left( {{C_{pg,i}} - R} \right)+\left( {1 - {\varepsilon _b}} \right){\rho _p}\mathop \sum \limits_{{i=1}}^{N} {q_i}\left( {{C_{pg,i}} - R} \right)} \right\}\frac{{\partial T}}{{\partial t}} \hfill \\ +{v_g}{\rho _g}\mathop \sum \limits_{{i=1}}^{N} {C_{pg,i}}\frac{{\partial T}}{{\partial z}}+\left( {1 - {\varepsilon _b}} \right){\varepsilon _p}\mathop \sum \limits_{{i=1}}^{N} {C_{pg,i}}\frac{{\partial {q_i}}}{{\partial t}}\Delta {H_i}+2h\frac{{T - {T_w}}}{{{R_b}}} \hfill \\ - \left( {{\varepsilon _b}+\left( {1 - {\varepsilon _b}} \right){\varepsilon _p}} \right)\frac{{\partial P}}{{\partial t}} - {k_g}\frac{{{\partial ^2}T}}{{\partial {z^2}}}=0 \hfill \\ \end{gathered}$
Gas phase momentum balance	$- \frac{{\partial {\text{P}}}}{{\partial {\text{z}}}}=\frac{{150{{{\upmu}}}{{\left( {1 - {{{{\upvarepsilon}}}_{\text{b}}}} \right)}^2}}}{{{{{\upvarepsilon}}}_{{\text{b}}}^{3}{{\left( {2{{\text{R}}_{\text{p}}}} \right)}^2}}}{{\text{v}}_{\text{g}}}+1.75\frac{{\left( {1 - {{{{\upvarepsilon}}}_{\text{b}}}} \right){{{{\uprho}}}_{\text{g}}}}}{{2{{\text{R}}_{\text{P}}}{{{\upvarepsilon}}}_{{\text{b}}}^{3}}}\left| {{{\text{v}}_{\text{g}}}} \right|{{\text{v}}_{\text{g}}}$
Langmuir isotherm	${q^{*}}=\frac{{{q_{m,1}}{b_i}{P_i}}}{{1+\mathop \sum \nolimits{b_i}{P_i}}}{b_i}=b_{i}^{0}exp\left( { - \frac{{\Delta {H_i}}}{{RT}}} \right)$
Linear driving force	$\frac{{\partial {q_i}}}{{\partial t}}=\frac{{15{D_{C,i}}}}{{R_{p}^{2}}}\left( {q_{i}^{{\text{*}}} - {q_i}} \right)$
Diffusion coefficient	${D_{ax}}=0.73{D_m}+\frac{{{v_g}{R_p}}}{{{\varepsilon _b}\left( {1+9.49\frac{{{\varepsilon _b}{D_m}}}{{2{v_g}{R_p}}}} \right)}}\quad{D_{k,i}}=48.5{D_p}\sqrt {\frac{T}{{{M_i}}}}$ $D_{m} = \frac{{0.01013T^{{1.75}} \sqrt {\left( {\frac{1}{{M_{A} }} + \frac{1}{{M_{B} }}} \right)} }}{{P\left( {D_{{V,A}}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} + D_{{V,B}}^{{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-\nulldelimiterspace} 3}}} } \right)}}\quad D_{{c,i}} = \frac{{\varepsilon _{p} }}{\tau }\frac{{D_{{k,i}} D_{m} }}{{D_{{k,i}} + D_{m} }}$
Boundary condition	${\text{Z}}=0:{v_{g,z}} \geqslant 0,\,{T_z}={T_{in}},\,{y_{i,z}}={y_{in,i}}\,else\frac{{\partial {T_z}}}{{\partial z}}=0,\,\frac{{\partial {y_{i,z}}}}{{\partial z}}=0$ ${\text{Z}}={\text{Hb}}:{v_{g,z}} \leqslant 0,\,{T_z}={T_{out}},\,{y_{i,z}}={y_{out,i}}\,else,\,\frac{{\partial {T_z}}}{{\partial z}}=0,\frac{{\partial {y_{i,z}}}}{{\partial z}}=0$

1.1 Sub-models

Sub-models in a VPSA process usually contain buffer tanks, valves and compressors. Dead space in this paper used the same equations as buffer tanks did. Detailed equations are listed as follows:

Buffer tank	Expression
Energy balance	$\frac{{\partial T\left( {\mathop \sum \nolimits^{} {n_i}C{p_i}} \right)}}{{\partial t}} - \mathop \sum \limits_{{i=1}}^{N} {F_i}{T_i}\mathop \sum \nolimits^{} {y_k}{C_{pg,i}}+\mathop \sum \limits_{{i=1}}^{N} {F_o}{T_o}\mathop \sum \nolimits^{} {y_k}{C_{pg,i}}=0$
Mass balance	$\frac{{\partial {n_i}}}{{\partial t}} - \mathop \sum \nolimits^{} {F_i}{y_i}+\mathop \sum \nolimits^{} {F_o}{y_o}=0$
Valve	${\text{if}}\,{P_{in}}>{P_{out}}F=\frac{{{C_v}\left( {{P_{in}} - {P_{out}}} \right)}}{{{{10}^6}}}\quad else\,F=0$
Compressor and vacuum	${\text{if}}\,{P_{out}}>{P_{in}}\,{\text{dW}}=\frac{{{P_{out}}{V_{in}}}}{{{\eta _p}}}\frac{\gamma }{{\gamma - 1}}\left[ {{{\left( {\frac{{{P_{out}}}}{{{P_{in}}}}} \right)}^{\frac{{\gamma - 1}}{\gamma }}} - 1} \right]$ ${\text{else}}\,{\text{dW}} =0\quad{W_{cycle}}=\mathop \smallint \limits_{0}^{{{t_{cycle}}}} dW$
Performance indicators	$purit{y_{C{O_2}}}=\frac{{\mathop \smallint \nolimits_{0}^{{{t_{cycle}}}} {F_{out}}{y_{out}}dt}}{{\mathop \smallint \nolimits_{0}^{{{t_{cycle}}}} {F_{out}}dt}}$ $recover{y_{C{O_2}}}=\frac{{\mathop \smallint \nolimits_{0}^{{{t_{cycle}}}} {F_{out}}{y_{out}}dt}}{{\mathop \smallint \nolimits_{0}^{{{t_{cycle}}}} {F_{in}}{y_{in}}dt}}$ $productivit{y_{C{O_2}}}=\frac{{3600\mathop \smallint \nolimits_{0}^{{{t_{cycle}}}} {F_{out}}{y_{out}}dt}}{{{t_{cycle}}{W_{ads}}}}$

Other data and values for all equations above are listed as follows and remain the same as our previous work. Langmuir non-isothermal model parameters for N₂ and CO₂ on silica gel were the curving fitting results of pure gas adsorption quantity on silica gel at different temperature and pressure.

Based on the mathematical models and parameters, the VPSA process is well established under fixed decision variables. In all states analyzed in this paper, all variables in Table 5 remains the same. VAD in table refers to the valve opening (Cv in valve mathematical model) of valve AD (in Fig. 1) and so does the same of other valves. FC1 refers to the flowrate of compressor C₁ (in Fig. 1) and so does C₂ and vacuum. Valve name that not appeared in the table but used in process means that these valves belongs to the kind of switch valve and do not have the function to adjust the flowrate.

Table 5 Operate decision variables used in the process